To determine the values of b when the equation has real solutions, we can use the discriminant in the Quadratic Formula.
x=b ±sqrt(b^2-4ac)/2a
The following applies for the number of roots in a second degree equation.
Complex Roots: b^2-4ac<0
One Real Root: b^2-4ac=0
Two Real Roots: b^2-4ac>0
As we can see, the second degree function has real roots as long as the discriminant is non-negative.
b^2-4ac≥0
By substituting the equation's value of a, b, and c into the inequality and solving for b, we can determine what values of b will result in either one or two roots.
Whenever the square of b is greater than or equal to 400, the equation will have real solutions. This will occur when b is less than or equal to - 20 or greater than or equal to 20. We get the following solutions set
b≤ - 20 or b≥ 20
